Mathematical modeling of the low and high affinity arabinose transport systems in Escherichia coli

Boolean Models of the Transport, Synthesis, and Metabolism of Tryptophan in Escherichia coli

The tryptophan (trp) operon in Escherichia coli codes for the proteins responsible for the synthesis of the amino acid tryptophan from chorismic acid, and has been one of the most well-studied gene networks since its discovery in the 1960s. The tryptophanase (tna) operon codes for proteins needed to transport and metabolize it. Both of these have been modeled individually with delay differential equations under the assumption of mass-action kinetics. Recent work has provided strong evidence for bistable behavior of the tna operon. The authors of Orozco-Gómez et al. (Sci Rep 9(1):5451, 2019) identified a medium range of tryptophan in which the system has two stable steady-states, and they reproduced these experimentally. In this paper, we will show how a Boolean model can capture this bistability. We will also develop and analyze a Boolean model of the trp operon. Finally, we will combine these two to create a single Boolean model of the transport, synthesis, and metabolism of tryptophan. In this amalgamated model, the bistability disappears, presumably reflecting the ability of the trp operon to produce tryptophan and drive the system toward homeostasis. All of these models have longer attractors that we call "artifacts of synchrony", which disappear in the asynchronous automata. This curiously matches the behavior of a recent Boolean model of the arabinose operon in E. coli, and we discuss some open-ended questions that arise along these lines.

Bistability and Asynchrony in a Boolean Model of the L-arabinose Operon in Escherichia coli

The lactose operon in Escherichia coli was the first known gene regulatory network, and it is frequently used as a prototype for new modeling paradigms. Historically, many of these modeling frameworks use differential equations. More recently, Stigler and Veliz-Cuba proposed a Boolean model that captures the bistability of the system and all of the biological steady states. In this paper, we model the well-known arabinose operon in E. coli with a Boolean network. This has several complex features not found in the lac operon, such as a protein that is both an activator and repressor, a DNA looping mechanism for gene repression, and the lack of inducer exclusion by glucose. For 11 out of 12 choices of initial conditions, we use computational algebra and Sage to verify that the state space contains a single fixed point that correctly matches the biology. The final initial condition, medium levels of arabinose and no glucose, successfully predicts the system's bistability. Finally, we compare the state space under synchronous and asynchronous update and see that the former has several artificial cycles that go away under a general asynchronous update.

Mathematical modeling reveals differential regulation of MAPK activity by phosphatase proteins in the yeast pheromone response pathway

To prevent indefinite cellular responses to external signals, cells utilize various adaptation mechanisms.

Bacterial sugar utilization gives rise to distinct single-cell behaviours

Inducible utilization pathways reflect widespread microbial strategies to uptake and consume sugars from the environment. Despite their broad importance and extensive characterization, little is known how these pathways naturally respond to their inducing sugar in individual cells. Here, we performed single-cell analyses to probe the behaviour of representative pathways in the model bacterium Escherichia coli. We observed diverse single-cell behaviours, including uniform responses (d-lactose, d-galactose, N-acetylglucosamine, N-acetylneuraminic acid), 'all-or-none' responses (d-xylose, l-rhamnose) and complex combinations thereof (l-arabinose, d-gluconate). Mathematical modelling and probing of genetically modified pathways revealed that the simple framework underlying these pathways - inducible transport and inducible catabolism - could give rise to most of these behaviours. Sugar catabolism was also an important feature, as disruption of catabolism eliminated tunable induction as well as enhanced memory of previous conditions. For instance, disruption of catabolism in pathways that respond to endogenously synthesized sugars led to full pathway induction even in the absence of exogenous sugar. Our findings demonstrate the remarkable flexibility of this simple biological framework, with direct implications for environmental adaptation and the engineering of synthetic utilization pathways as titratable expression systems and for metabolic engineering.

Dynamics matter: differences and similarities between alternatively designed mechanisms

Cells selectively respond to external stimuli to maintain cellular homeostasis by making use of different regulatory mechanisms. We studied two classes of signal-dependent regulatory inhibition and activation mechanisms in this study. Inhibition mechanisms assume that inhibition can occur in two different ways: either by increasing the degradation rate or decreasing the production rate. Similarly, it is assumed that signal-triggered activation can occur either through increasing production rate or decreasing degradation rate. We devised mathematical models (deterministic and stochastic) to compare and contrast responses of these activation and inhibition mechanisms to a time dependent discrete signal. Our simulation results show that the signal-dependent increased degradation mechanism is a more effective, noisier and quicker way to inhibit the protein abundance compared to the signal-dependent decreased activation mechanism. On the other hand, the signal-dependent increased production mechanism can produce a much stronger and faster response than the signal-dependent decreased degradation mechanism. However, our simulations predict that both of the activation mechanisms have roughly similar noise structures. Our analysis exemplifies the importance of mathematical modeling in the analysis of biological regulatory networks.